These constructions are very different. The meanings are different, and as far as I know you could not substitute words that work with one into the other and make any sense:
I am too tired not to fall asleep if I lie down.
This means I will fall asleep if I lie down because I am so tired.
I would be only too happy to oblige.
This means I would very much enjoy doing as you have asked.
"Only too ... to" is of limited use and is pretty much a fixed expression. "Only" does not really make sense in front of "too", but the meaning of "only too" in this expression is something like "quite" (an understated intensifier).
Do you have sample sentences where these two constructions might have similar meanings?
These are interesting examples. I didn't get a chance to read them before I posted. I would agree with Forero that changing the construction in the first example to "
I am only too tired to fall asleep if I lie down" makes no sense to me.
In the second example, though,
I would be too happy not to oblige. Would imply that you would be
so happy that you would be unable to say no to someone's request. On the other hand, if it was changed to "I would be
all to happy not to oblige," it would mean exactly the opposite.
I have no idea how we account for these differences in meaning.